Long lines in subsets of large measure in high dimension
成果类型：
Article
署名作者：
Elboim, Dor; Klartag, Bo'az
署名单位：
Institute for Advanced Study - USA; Weizmann Institute of Science
刊物名称：
PROBABILITY THEORY AND RELATED FIELDS
ISSN/ISSBN：
0178-8051
DOI：
10.1007/s00440-023-01231-7
发表日期：
2023
页码：
657-695
关键词：
摘要：
We show that for any set A subset of [0, 1](n) with Vol(A) >= 1/2 there exists a line L such that the one-dimensional Lebesgue measure of L boolean AND A is at least Omega (n(1/4)). The exponent 1/4 is tight. More generally, for a probability measure mu on R-n and 0 < a < 1 define L(mu, a) := inf(A; mu(A)=a) sup(line) vertical bar L boolean AND A vertical bar where vertical bar center dot vertical bar stands for the one-dimensional Lebesgue measure. We study the asymptotic behavior of L(mu, a) when mu is a product measure and when mu is the uniform measure on the l(p) ball. We observe a rather unified behavior in a large class of product measures. On the other hand, for l(p) balls with 1 <= p <= infinity we find that there are phase transitions of different types.
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